Elliptic Integrals in Kerr Spacetime: Explicit Reduction, AGM Evaluation, and Photon-Sphere Universality

The geodesic structure of Kerr spacetime, within the class of observables expressible as separated Hamilton-Jacobi integrals with rational prefactors, reduces exactly—not approximately—to the three classical complete elliptic integrals K(k), E(k), and Π(n, k).

All three are evaluated via the Gauss-Legendre arithmetic-geometric mean (AGM), which converges quadratically: four to seven iterations suffice for 50-decimal precision across all physical Kerr moduli, with mathematically certified residuals at every step.

This framework yields exact closed-form reductions for eleven Kerr observables, including the shadow radius, photon-ring quantities, plasma lensing corrections, horizon geometry, orbital periods, and relativistic precessions. The remaining three quantities considered in this paper are either purely algebraic (horizon area, Hawking temperature, ISCO radius) or belong to a separate spectral extension treated approximately: the quasinormal-mode spectrum is modelled through the Ferrari-Mashhoon Pöschl-Teller approximation, with 1–9% errors on low overtones, while the asymptotic regime is recovered exactly through the Stirling-Ramanujan expansion of log Γ(z).

Three theorems establish the universality of the vertical extent of the Kerr photon sphere. In the exact geodesic sector, the logarithmic divergences at photon capture and ISCO instability are shown to arise from a common algebraic mechanism, namely root coalescence in the relevant Kerr polynomial potentials.

Version 2 note: 
Revised abstract and conclusion, clarified the distinction between the exact geodesic framework and the approximate spectral extension related to quasinormal modes and the neutron-star normal-mode transition, and added a working conjecture for companion work.